scholarly journals An issue based power index

2020 ◽  
Author(s):  
Qianqian Kong ◽  
Hans Peters
Keyword(s):  
Linear Order ◽  
Power Index ◽  
Simple Game ◽  
Power Indices ◽  
Weighted Sum ◽  
Simple Games ◽  
Linear Orders ◽  
Transfer Property ◽  
Power Change ◽  
Equal Power

Abstract An issue game is a combination of a monotonic simple game and an issue profile. An issue profile is a profile of linear orders on the player set, one for each issue within the set of issues: such a linear order is interpreted as the order in which the players will support the issue under consideration. A power index assigns to each player in an issue game a nonnegative number, where these numbers sum up to one. We consider a class of power indices, characterized by weight vectors on the set of issues. A power index in this class assigns to each player the weighted sum of the issues for which that player is pivotal. A player is pivotal for an issue if that player is a pivotal player in the coalition consisting of all players preceding that player in the linear order associated with that issue. We present several axiomatic characterizations of this class of power indices. The first characterization is based on two axioms: one says how power depends on the issues under consideration (Issue Dependence), and the other one concerns the consequences, for power, of splitting players into several new players (no advantageous splitting). The second characterization uses a stronger version of Issue Dependence, and an axiom about symmetric players (Invariance with respect to Symmetric Players). The third characterization is based on a variation on the transfer property for values of simple games (Equal Power Change), besides Invariance with respect to Symmetric Players and another version of Issue Dependence. Finally, we discuss how an issue profile may arise from preferences of players about issues.

scholarly journals An Axiomatization for Two Power Indices for (3,2)-Simple Games

2019 ◽  
Vol 21 (01) ◽  
pp. 1940001 ◽  
Author(s):  
Giulia Bernardi ◽  
Josep Freixas
Keyword(s):  
Power Index ◽  
Power Indices ◽  
Simple Games ◽  
Banzhaf Index ◽  
Null Player

The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

Two results on borel orders

1989 ◽  
Vol 54 (3) ◽  
pp. 865-874 ◽  
Author(s):  
Alain Louveau
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Lexicographic Ordering ◽  
Countable Ordinal

AbstractWe prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

scholarly journals Power Indices in the Context of Social Learning Behaviour in Social Networks

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Ruili Shi ◽  
Chunxiang Guo ◽  
Xin Gu
Keyword(s):  
Social Networks ◽  
Social Network ◽  
Social Learning ◽  
Social Impact ◽  
Power Index ◽  
Decision Makers ◽  
Power Indices ◽  
Learning Efficiency ◽  
Relationship Closeness ◽  
The Social

This paper puts forward the concept of integrated power, synthetically measures the voters’ ability to influence the results of decision-making by influencing others through social learning, considering the interactions between decision-makers in social networks, and offers a method for measuring integrated power. Based on the theory and model of social learning, we analyze the influence of social learning on the voting process and power indices from the perspective of individuals’ professional level, position within the social network structure, relationship closeness, and learning efficiency. A measurement model of integrated power is constructed, and the variation in integrated power compared with that of the Banzhaf index is analyzed by numerical simulation. The results show that when the individual’s professional level is higher and closeness with neighboring decision-makers is greater, then the integrated power index is higher. An individual’s integrated power index may decrease when he/she changes from an isolated node to a nonisolated node, and then his/her integrated power will increase with the increases of neighbor nodes. Social learning efficiency can promote the integrated power of individuals with lower social impact and relationship closeness, but it is not beneficial for the core and influential members of the social network.

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

NEW APPORTIONMENT METHODS USING SIMPLE GAMES

2010 ◽  
Vol 12 (03) ◽  
pp. 211-222
Author(s):  
EVAN SHELLSHEAR
Keyword(s):  
Simple Game ◽  
Simple Games ◽  
New Methods ◽  
Apportionment Methods

This paper investigates the suitability of new apportionment methods based on the idea of preserving the coalition function of the simple game generated by the populations of the states of some country. The new methods fill a gap in the literature concerning apportionment methods based on winning coalitions. The main results in this paper show that the new apportionment methods do not satisfy desirable properties such as house monotonicity, quota, etc.

scholarly journals On cuts in ultraproducts of linear orders I

2016 ◽  
Vol 16 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Mohammad Golshani ◽  
Saharon Shelah
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Regular Cardinals

For an ultrafilter [Formula: see text] on a cardinal [Formula: see text] we wonder for which pair [Formula: see text] of regular cardinals, we have: for any [Formula: see text]-saturated dense linear order [Formula: see text] has a cut of cofinality [Formula: see text] We deal mainly with the case [Formula: see text]

scholarly journals Bounds for Owen's Multilinear Extension

2007 ◽  
Vol 44 (4) ◽  
pp. 852-864 ◽  
Author(s):  
Josep Freixas
Keyword(s):  
Game Theory ◽  
Simple Game ◽  
Practical Interest ◽  
Simple Games ◽  
Theoretical Interest ◽  
Complex Component ◽  
Multilinear Extension ◽  
Voting System ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

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